Principles and psychophysics of Active Inference - Probabilities and Optimal Inference to Understand the Brain

Principles and psychophysics of Active Inference in anticipating a dynamic probabilistic bias

Laurent Perrinet, Chloé Pasturel and Anna Montagnini

Probabilities and Optimal Inference to Understand the Brain, 5/4/2018

Acknowledgements:

Jim Bednar @ Continuum Analytics & University of Edinburgh
Berk Mirza, Rick Adams and Karl Friston @ UCL - Wellcome Trust Centre for Neuroimaging
Wahiba Taouali, Giacomo Benvenuti, Frédéric Chavane - ANR BalaV1
Jean-Bernard Damasse, Laurent Madelain - ANR REM

http://invibe.net/LaurentPerrinet/Presentations/2018-04-05_BCP_talk

Principles and psychophysics of Active Inference in anticipating a dynamic probabilistic bias

Laurent Perrinet, Chloé Pasturel and Anna Montagnini

Probabilities and Optimal Inference to Understand the Brain, 5/4/2018

Acknowledgements:

Jim Bednar @ Continuum Analytics & University of Edinburgh
Berk Mirza, Rick Adams and Karl Friston @ UCL - Wellcome Trust Centre for Neuroimaging
Wahiba Taouali, Giacomo Benvenuti, Frédéric Chavane - ANR BalaV1
Jean-Bernard Damasse, Laurent Madelain - ANR REM

Outline

Motivation

Raw psychophysical results

The Bayesian Changepoint Detector

Results using the BCP

Motivation - a Real-life example

Motivation - Eye Movements

Montagnini A, Souto D, and Masson GS (2010) J Vis (VSS Abstracts) 10(7):554,
Montagnini A, Perrinet L, and Masson GS (2015) BICV book chapter

Motivation - Eye Movements

Motivation - Random-length block design

Outline

Motivation

Raw psychophysical results

The Bayesian Changepoint Detector

Results using the BCP

Raw psychophysical results - Random-length block design

full code @ github.com/chloepasturel/AnticipatorySPEM

Raw psychophysical results

First, we overlay the results of the bet result for one of the 12 subjects

We rescaled th value given by the observer so that it fits to 0 (sure it goes left) to 1 (sure it goes right)

We observe a pretty good fit of this trace as a function of trial number

In particular, we see 2 main effects: - results are more variable when the bias is around .5 than when it is high (close to zero or one) - switches were detected quite rapidly but with a certain delay of a few trials. Indeed, note that this plot shows the entire sequence but that observers had only access to some internal representation of the memory of the previous observations. When faced with some new observations, the observer has to constantly adapt his response to either exploit this information by considering that this observation belongs to the same sub-block or to explore a novel hypothesis. This compromise is one of the crucial component that we wish to explore.

Let's now have a look at EMs...

Raw psychophysical results - Fitting eye movements

I show here a typical velocity traces for one subject / 2 trials

x-axis is time in milliseconds aligned on target onset, and we show respectively from left to right the fixation in gray, the GAP in pink (300ms) and the run in light gray.
y-axis is the velocity as computed as the gradient of position. Remark that the eyelink provides with the periods of saccades or blinks that we removed from the signal. it is quite noisy and to complement existing signal processing methods, Chloe implemented a robust
fitting method which allows to extract some key components of the velocity traces: maximum speed, latency, temporal inertia ($ au$) and most interestingly acceleration before motion onset. We cross-validated that this method was givinfg similar results to other classical methods but in a more robust fashion/

While being sensible to recording errors, this allows us to extract the anticipatory component of SPEMs and..

Raw psychophysical results - Fitting eye movements

I show here a typical velocity traces for one subject / 2 trials

x-axis is time in milliseconds aligned on target onset, and we show respectively from left to right the fixation in gray, the GAP in pink (300ms) and the run in light gray.
y-axis is the velocity as computed as the gradient of position. Remark that the eyelink provides with the periods of saccades or blinks that we removed from the signal. it is quite noisy and to complement existing signal processing methods, Chloe implemented a robust
fitting method which allows to extract some key components of the velocity traces: maximum speed, latency, temporal inertia ($ au$) and most interestingly acceleration before motion onset. We cross-validated that this method was givinfg similar results to other classical methods but in a more robust fashion/

While being sensible to recording errors, this allows us to extract the anticipatory component of SPEMs and..

Raw psychophysical results

quantitatively, one can now plot the results for all subjects
the x-axis corresponds to the probability that was coded at the second layer and which is unknown to the observer
the y -axis shows either the bet or the
dots represent single responses - the saturation giving the identity of the observer

we notice a quite nice linear correlation (black line) for both experiments; of the order of that found in the classical experiment with fixed blocks and a vartiety of bias values. This is surprising as the blocks are of random length, observer can still adapt to such a volatile environment.

Another visualization, the scatter plot of acceleration as a function of probability bet shows also that there is a correlation between both variables.

This allows to make a first point: it is possible to use more genreal models such as hierarchical generative models.

However, while this results seem encouraging, a more finer analysis may be necessary.

Raw psychophysical results

quantitatively, one can now plot the results for all subjects
the x-axis corresponds to the probability that was coded at the second layer and which is unknown to the observer
the y -axis shows either the bet or the
dots represent single responses - the saturation giving the identity of the observer

Another visualization, the scatter plot of acceleration as a function of probability bet shows also that there is a correlation between both variables.

This allows to make a first point: it is possible to use more genreal models such as hierarchical generative models.

However, while this results seem encouraging, a more finer analysis may be necessary.

Outline

Motivation

Raw psychophysical results

The Bayesian Changepoint Detector

Results using the BCP

The Bayesian Changepoint Detector

Let's remember our hierarchical generative model.

At any given trial, we wish to construct an algorithm which

We will introduce a fundamental component of Bayesian models : a latent variable

this new variable will be used to test different hypothesis which will be evaluated to predict future states. it is called latent because it aims at representing a variable that is latent (or hidden) to the observer

in our case, we will assume that the bayesian model knows about the structure of the generative model and we will set it to the current run-length $r$, that is, at any given trial, the hypothesis that the past r observations belong to the same block. of course a wrong choice of a latent variables (let's say the temperture in the experimental room) may give unexpected results, even is the bayesian model is "optimal" - an essential point to understand in bayesian inference

The Bayesian Changepoint Detector

Initialize $P(r_0=0)=1$ and $ν^{(0)}_1 = ν_{prior}$ and $χ^{(0)}_1 = χ_{prior}$

The Bayesian Changepoint Detector

Observe New Datum $x_t$ and Perform Prediction $P (x_{t+1} | x_{1:t}) = P (x_{t+1}|x_{1:t} , r_t) \cdot P (r_t|x_{1:t})$

The Bayesian Changepoint Detector

Evaluate Predictive Probability $π_{1:t} = P(x_t |ν^{(r)}_t,χ^{(r)}_t)$
Calculate Growth Probabilities $P(r_t=r_{t-1}+1, x_{1:t}) = P(r_{t-1}, x_{1:t-1}) \cdot π^{(r)}_t \cdot (1−H(r^{(r)}_{t-1}))$
Calculate Changepoint Probabilities $P(r_t=0, x_{1:t})= \sum_{r_{t-1}} P(r_{t-1}, x_{1:t-1}) \cdot π^{(r)}_t \cdot H(r^{(r)}_{t-1})$

The Bayesian Changepoint Detector

Calculate Evidence $P(x_{1:t}) = \sum_{r_{t-1}} P (r_t, x_{1:t})$
Determine Run Length Distribution $P (r_t | x_{1:t}) = P (r_t, x_{1:t})/P (x_{1:t}) $

The Bayesian Changepoint Detector

Update Sufficient Statistics : $ν^{(r+1)}_{t+1} = ν^{(r)}_{t} +1$, $χ^{(r+1)}_{t+1} = χ^{(r)}_{t} + u(x_t)$
$ν^{(0)}_{t+1} = ν_{prior}$, $χ^{(0)}_{t+1} = χ_{prior}$

Bayesian Changepoint Detector

Initialize

$P(r_0)= S(r)$ or $P(r_0=0)=1$ and
$ν^{(0)}_1 = ν_{prior}$ and $χ^{(0)}_1 = χ_{prior}$

Observe New Datum $x_t$
Evaluate Predictive Probability $π_{1:t} = P(x_t |ν^{(r)}_t,χ^{(r)}_t)$
Calculate Growth Probabilities $P(r_t=r_{t-1}+1, x_{1:t}) = P(r_{t-1}, x_{1:t-1}) \cdot π^{(r)}_t \cdot (1−H(r^{(r)}_{t-1}))$
Calculate Changepoint Probabilities $P(r_t=0, x_{1:t})= \sum_{r_{t-1}} P(r_{t-1}, x_{1:t-1}) \cdot π^{(r)}_t \cdot H(r^{(r)}_{t-1})$
Calculate Evidence $P(x_{1:t}) = \sum_{r_{t-1}} P (r_t, x_{1:t})$
Determine Run Length Distribution $P (r_t | x_{1:t}) = P (r_t, x_{1:t})/P (x_{1:t}) $
Update Sufficient Statistics :

$ν^{(0)}_{t+1} = ν_{prior}$, $χ^{(0)}_{t+1} = χ_{prior}$
$ν^{(r+1)}_{t+1} = ν^{(r)}_{t} +1$, $χ^{(r+1)}_{t+1} = χ^{(r)}_{t} + u(x_t)$

Perform Prediction $P (x_{t+1} | x_{1:t}) = P (x_{t+1}|x_{1:t} , r_t) \cdot P (r_t|x_{1:t})$
go to (2)

The Bayesian Changepoint Detector

Outline

Motivation

Raw psychophysical results

The Bayesian Changepoint Detector

Results using the BCP

Results using the BCP - inference with BCP

Results using the BCP - fit with BCP

Principles and psychophysics of Active Inference in anticipating a dynamic probabilistic bias

Laurent Perrinet, Chloé Pasturel and Anna Montagnini

Probabilities and Optimal Inference to Understand the Brain, 5/4/2018

Acknowledgements:

Jim Bednar @ Continuum Analytics & University of Edinburgh
Berk Mirza, Rick Adams and Karl Friston @ UCL - Wellcome Trust Centre for Neuroimaging
Wahiba Taouali, Giacomo Benvenuti, Frédéric Chavane - ANR BalaV1
Jean-Bernard Damasse, Laurent Madelain - ANR REM